Problem 22 - Algebra and Trigonometry

Question

An initial amplitude \(k,\) damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p . )\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(17-20,\) and of the form \(y=k e^{-c t}\) sin \(\omega t\) in Exercises \(21-24\). (b) Graph the function. $$k=1, \quad c=1, \quad p=1$$

Step-by-Step Solution

Answer

The function is \( y = e^{-t} \cos(2\pi t) \).

Step 1: Determine the Angular Frequency

The period \( p \) is given as \( p = 1 \). Since frequency \( f \) and period \( p \) are related by \( f = \frac{1}{p} \), we can first find the frequency as \( f = \frac{1}{1} = 1 \). The angular frequency \( \omega \) is related to the frequency by the formula \( \omega = 2\pi f \). Thus, \( \omega = 2\pi \times 1 = 2\pi \).

Step 2: Write the Function Equation

Since \( k = 1 \), \( c = 1 \), and \( \omega = 2\pi \), we can write the function for damped harmonic motion. According to Exercise \( 17-20 \), we use the form: \[ y = k e^{-ct} \cos(\omega t) \]Substitute the values:\[ y = 1 \cdot e^{-1 \cdot t} \cdot \cos(2\pi t) \]So, the function is:\[ y = e^{-t} \cos(2\pi t) \]

Step 3: Graph the Function

To graph the function \( y = e^{-t} \cos(2\pi t) \), plot it over a suitable range of \( t \). Note that this function represents a cosine wave with an exponentially decaying amplitude due to the \( e^{-t} \) term. At \( t = 0 \), \( y = \cos(0) = 1 \). As \( t \) increases, the amplitude diminishes because of \( e^{-t} \), and the function will oscillate between decreasing positive and negative values. The wave completes one full cycle when \( t = \frac{1}{f} = 1 \).

Key Concepts

Angular Frequency

Angular frequency is a critical component of understanding harmonic motion. It tells us how fast the oscillation cycles repeat themselves. The angular frequency is denoted by the symbol \( \omega \) and is given by the formula \( \omega = 2\pi f \), where \( f \) is the frequency.

In the exercise, with a period \( p \) of 1, the frequency \( f \) becomes \( f = \frac{1}{p} = 1 \). With this frequency, the angular frequency becomes \( \omega = 2\pi \times 1 = 2\pi \).

• It is measured in radians per second.
• Provides insight into the rapidity of cycles within one full turn of a circle.
• In harmonic motion, helps relate time-based cycles to angular cycles.

This concept is foundational to relating the periodic nature of waveforms with the circular nature of oscillations.

Exponential Decay

Exponential decay describes how the amplitude of the wave decreases over time. It is characterized by the term \( e^{-ct} \) in the damped harmonic motion equation. Here, \( c \) is the damping constant, which shows how quickly the motion decays.

The form \( e^{-ct} \) represents a mathematical function where:

• With time \( t \) increasing, the \( e^{-ct} \) factor makes the amplitude smaller.
• The damping constant \( c \) determines how quickly the amplitude drops. Greater \( c \) values lead to faster decay.

In our example, with \( c = 1 \), the term \( e^{-t} \) indicates that the motion's amplitude halves at every unit time increase, making the oscillations fade gradually.

Cosine Function

The cosine function, represented as \( \cos(\omega t) \), is pivotal in calculating the wave's position at any given point in time. In harmonic motion, it helps in understanding how waves shift between maximum and minimum positions.

The cosine function is typically used because:

• It starts at maximum value when \( t=0 \), making it ideal for oscillations that begin at a peak.
• It reflects how the wave position is constantly changing in a smooth, periodic pattern.
• With angular frequency included, \( 2\pi \) radians complete a full wave cycle.

In the context of our problem, \( \cos(2\pi t) \) provides the periodic element that governs where the wave is in its cycle as time progresses. This oscillation is overlaid with the decaying amplitude from the exponential component.

Oscillation Period

The oscillation period is the duration it takes for the wave to complete a full cycle. It's the inverse of frequency and is denoted by \( p \). This parameter is crucial for understanding time intervals over which the cycle of motion repeats.

Concisely, it has these characteristics:

• The period is defined as \( p = \frac{1}{f} \).
• In our exercise, \( p = 1 \) indicates every second introduces a full wave repeat.
• The period helps determine how synchronized the motion is over extended intervals.

When analyzing graphs of harmonic motion, recognizing the period highlights cycles' timing and anticipates repeated patterns in oscillations. The consistent periodic timing provides a rhythm to the entire harmonic motion process.